Kinematics
Motion in a straight line and in a plane.
Position, Displacement, Distance
Vector vs scalar quantities, frame of reference.
Velocity and Acceleration
Average vs instantaneous, units, sign conventions.
Equations of Motion (1D)
v = u + at, s = ut + ½at², v² = u² + 2as.
When an object moves with uniform acceleration (a = constant) in a straight line, three equations describe its motion. Let u = initial velocity, v = velocity after time t, s = displacement, a = acceleration.
First equation (v in terms of t): v = u + at
This comes directly from the definition a = (v − u) / t.Second equation (s in terms of t): s = ut + ½at²
Derived by integrating velocity, or geometrically as the area under a v–t graph (a trapezium).Third equation (v in terms of s): v² = u² + 2as
Useful when time is unknown — comes from eliminating t between the first two.
Sign convention: choose one direction as positive. Acceleration opposite to velocity is negative (deceleration).
Worked example. A car starts from rest and accelerates at 2 m/s² for 5 s. Its final velocity is v = 0 + 2 × 5 = 10 m/s, and the distance covered is s = 0 + ½ × 2 × 5² = 25 m.
Mistake 1: Using these for non-uniform acceleration. The three equations only work when a is constant. If acceleration varies, you must integrate.
Mistake 2: Forgetting that displacement and distance are different. Displacement is a vector (can be negative); distance is the total path length (always positive). For a ball thrown up and caught back at the same height, distance > 0 but displacement = 0.
Mistake 3: Mixing up reference directions. If you take "up" as positive, then g = −9.8 m/s², not +9.8.
Mistake 4: Plugging in average velocity for u or v. u and v are the velocities at the start and end of the interval, not somewhere in between.
Projectile Motion
Range, max height, time of flight derivations.
For a particle launched with speed u at angle θ above the horizontal (with no air resistance, on flat ground):
• Time of flight: T = (2u sin θ) / g
• Maximum height: H = (u² sin²θ) / (2g)
• Horizontal range: R = (u² sin 2θ) / g
• Velocity at any time t: v = √( (u cos θ)² + (u sin θ − gt)² )
Key insight: the range R is maximized when 2θ = 90°, i.e. when θ = 45°. The maximum range is R_max = u² / g.
Two angles that give the same range: θ and (90° − θ). For example, 30° and 60° give the same horizontal distance (but different heights and times).